1893.] the Mathematical Theory of Evolution. 331 



asymmetrical curve, we can proceed as follows : Imagine n to be 

 large, but the ratio q[p either small or large ; then we can obtain 

 generalised form of the normal curve of an asymmetrical character; 

 its equation referred to the centroid vertical is 



r y^^n / j 

 1 rG8+i) A + */-* 



where y3 stands for 4 /< 3 3 //t 2 3 and F(/5) is the Ealerian gamma function.* 

 Putting ytt 3 = for an asymmetrical curve, the equation takes an in- 

 determinate form obtained by putting ft = co, but on evaluation we 

 have the usual normal form : 



This generalised probability curve fits with a high degree of 

 accuracy a number of measurements and observations hitherto not 

 reduced to theoretical treatment, e.g., barometric frequency curves. 



The importance of this first dissection of asymmetrical frequency 

 curves lies in the fact that it measures the theoretical number n of 

 contributory " causes " and the odds p : q that an element of devia- 

 tion will be positive. The whole theory is, however, of an elementary 

 character, and, as biological frequency curves often tend to develop a 

 double-humped character,"f they do not invariably fall under this 

 class, and it is not dealt with at length in the memoir. 



5. Class b. The general theory of the dissection of a given 

 abnormal frequency curve into m components is not dealt with, partly 

 on account of its exceedingly great analytical difficulties, partly 

 because there is an a priori probability that we have a mixture of 

 only two homogeneous groups, or from the standpoint of evolution 

 that the species will break up at first into two, rather than three or 

 more, families. At any rate, the dissection into two is likely to give 

 us either the chief components or a measure of the chief asymmetry 

 of the curve. Supposing the curve asymmetrical, it is shown that the 

 solution of the problem is theoretically unique, but it is pointed out 

 that in statistical practice our curve is based upon a limited number 

 of measurements, and is therefore not an accurately true compound 

 of two normal groups. A theoretical test is given to distinguish 

 between the better of two or more solutions. The method adopted 

 for the dissection is based on equality of the first five moments and 

 of the areas of the abnormal curve and of its two components. This 

 method is j ustified in the same manner as the determination of the 



* If /S be large, it may be taken as approximately whole, and the factor in round 

 brackets is then unity. 



t E-g-) claspers of earwigs, height of Italian recruits of various special provinces, 

 short sight of Marlborough boys, height of inhabitants of Doubs, &c. 



VOL. LIV. 2 A 



